Duke Mathematical Journal

Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices

François Maucourant

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Abstract

We show that Haar measures of connected semisimple groups, embedded via a representation into a matrix space, have a homogeneous asymptotic limit when viewed from far away and appropriately rescaled. This is still true if the Haar measure of the semisimple group is replaced by the Haar measure of an irreducible lattice of the group, and the asymptotic measure is the same. In the case of an almost simple group of rank greater than 2, a remainder term is also obtained. This extends and makes precise anterior results of Duke, Rudnick, and Sarnak [DRS] and Eskin and McMullen [EM] in the case of a group variety

Article information

Source
Duke Math. J., Volume 136, Number 2 (2007), 357-399.

Dates
First available in Project Euclid: 21 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1166711374

Digital Object Identifier
doi:10.1215/S0012-7094-07-13626-3

Mathematical Reviews number (MathSciNet)
MR2286635

Zentralblatt MATH identifier
1117.22006

Subjects
Primary: 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05} 11P21: Lattice points in specified regions
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]

Citation

Maucourant, François. Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices. Duke Math. J. 136 (2007), no. 2, 357--399. doi:10.1215/S0012-7094-07-13626-3. https://projecteuclid.org/euclid.dmj/1166711374


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